# Reducing the Magnesium Content from Seawater to Improve Tailing Flocculation: Description by Population Balance Models

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}.

## 2. Methodology

#### 2.1. Materials

^{+}, 1.38 g/L Mg

^{2+}, 0.4 g/L Ca

^{2+}, 0.39 K

^{+}, 19.6 Cl

^{−}, and 0.15 g/L HCO

_{3}

^{−}[41].

_{2}Si

_{2}O

_{5}(OH)

_{4}) and 16 wt% halloysite (Al

_{2}Si

_{2}O

_{5}(OH)

_{4}·2H

_{2}O) (Figure 1). A D5000 X-ray diffractometer (Siemens S.A., Lac Condes, Chile) was used and the data were processed with Total Pattern Analysis Software (TOPAS) (Siemens S.A., Lac Condes, Chile). Quartz was acquired from a local Chilean store, where the SiO

_{2}content detected by quantitative XRD was over 99 wt% (see Figure 2). Both quartz and kaolin had a density of 2.6 g/t. A Microtrac S3500 laser diffraction particle size analyzer (Verder Scientific, Newtown, PA, USA) was used. The analysis showed that 10% of the particles were smaller than d10 = 1.8 and 3.8 µm in the kaolin and quartz samples, respectively.

#### 2.2. Magnesium Removal

^{+}, 0.01 g/L Mg

^{2+}, 2.35 g/L Ca

^{2+}, 0.39 K

^{+}, 19.6 Cl

^{−}, and 0.05 g/L HCO

_{3}.

#### 2.3. Flocculant-Suspension

#### 2.4. Batch Settling Tests

^{3}cylinders (35 mm internal diameter), and then slowly inverting the cylinder two times by hand (the whole cylinder rotation process took, in all cases, about 4 s). After 10 min of settling, the supernatant fluid was rescued and stirred in order to homogenize the suspended solids. Then, a 50 mL aliquot was used for turbidity measurements in a HANNA HI98713 turbidimeter (Hanna Instruments, Santiago, Chile), which performed ten readings in 20 s, delivering the average at the end of that period.

#### 2.5. Characterization of Aggregates

## 3. Modeling

_{i}

_{+1}) = 2V

_{i}). The PBM equation is given by:

- The first and second terms describe the aggregate formation of size i from smaller aggregates.
- The third and fourth terms describe the aggregation death of size i to higher aggregates.
- The fifth term represents the breakage formation of size i from the rupture of a greater aggregate.
- The sixth term represents the breakage death of size i by creating smaller aggregates.

#### 3.1. Aggregation Kernel

_{i}is the permeability; we use the expression from the work by Li and Logan [47]:

#### 3.2. Breakage Kernel

#### 3.3. Shear Rate

#### 3.4. Solution

## 4. Results

#### 4.1. Input Parameters and Distribution

#### 4.2. Flocculation Kinetics and Modeling

#### 4.3. Optimized Parameters

_{1}is small, the ${a}_{min}$ is also small. This trend must be followed in order to satisfy steady-state conditions in the aggregation process. The s

_{2}parameters show values between 1 and 2, but as the s

_{1}parameter is small, its contribution is neglected from the global behavior. From this, we see that high dosage increments contribute to the aggregate kernel rather than the breakage kernel.

#### 4.4. Aggregation, Breakage, and Permeability Modeling

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**(

**A**) Normalized initial volume distribution of particles of synthetic tailings in seawater at a mixing rate of 150 rpm. (

**B**) Fractal dimension from the settling experiments.

**Figure 4.**Flocculation kinetics of synthetic tailings in seawater as a function of flocculant doses (mixing rate 150 rpm). Solid circles correspond to experimental data and solid lines to the best fit with the population balance model (PBM). (

**A**) Seawater (SW) pH 8. (

**B**) SW pH 11. (

**C**) Treated (T)-SW pH 11.

**Figure 5.**Optimum aggregation parameters vs. shear rate for constant and variable fractal dimension: (

**A**) Maximum and (

**B**) minimum collision efficiency and (

**C**) collision efficiency decay constant ${k}_{d}$.

**Figure 6.**Optimum breakage parameters vs. shear rate for constant and variable fractal dimension: (

**A**) S1 and (

**B**) collision efficiency decay constant ${k}_{d}$.

**Figure 7.**Collision frequency for the experimental data for (

**A**) SW pH 8, (

**B**) SW pH 11, and (

**C**) T-SW pH 11.

**Figure 8.**Collision efficiency for the experimental data for (

**A**) SW pH 8, (

**B**) SW pH 11, and (

**C**) T-SW pH 11.

**Figure 9.**Breakage rate for the experimental data for (

**A**) SW pH 8, (

**B**) SW pH 11, and (

**C**) T-SW pH 11.

**Figure 10.**Permeability for the experimental data for (

**A**) SW pH 8, (

**B**) SW pH 11, and (

**C**) T-SW pH 11.

${i}_{max}$ | $30$ | |

$\varphi $ | $0.054$ | |

$c$ | $0.65$ | |

${N}_{p}$ | $0.6$ | |

$D$ | $8.0$ | $\mathrm{cm}$ |

$V$ | $0.25$ | $\mathrm{l}$ |

${\rho}_{s}$ | $2600$ | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

${\rho}_{w}$ | $1000$ | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

${\mu}_{sus}$ | $0.005$ | $\mathrm{kg}/\left(\mathrm{ms}\right)$ |

$w$ | $0.08$ | |

d_{0} | $0.0005$ | $\mathrm{cm}$ |

System | Flocculant Dose, g/ton | GoF, % | R^{2} |
---|---|---|---|

SW pH 8 | 13 | 86.1 | 0.675 |

21 | 92.9 | 0.915 | |

28 | 93.8 | 0.939 | |

SW pH 11 | 13 | 89.9 | 0.629 |

21 | 89.6 | 0.706 | |

28 | 91.7 | 0.794 | |

T-SW pH 11 | 13 | 91.4 | 0.838 |

21 | 95.3 | 0.961 | |

28 | 94.9 | 0.956 |

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**MDPI and ACS Style**

Quezada, G.R.; Jeldres, M.; Toro, N.; Robles, P.; Jeldres, R.I. Reducing the Magnesium Content from Seawater to Improve Tailing Flocculation: Description by Population Balance Models. *Metals* **2020**, *10*, 329.
https://doi.org/10.3390/met10030329

**AMA Style**

Quezada GR, Jeldres M, Toro N, Robles P, Jeldres RI. Reducing the Magnesium Content from Seawater to Improve Tailing Flocculation: Description by Population Balance Models. *Metals*. 2020; 10(3):329.
https://doi.org/10.3390/met10030329

**Chicago/Turabian Style**

Quezada, Gonzalo R., Matías Jeldres, Norman Toro, Pedro Robles, and Ricardo I. Jeldres. 2020. "Reducing the Magnesium Content from Seawater to Improve Tailing Flocculation: Description by Population Balance Models" *Metals* 10, no. 3: 329.
https://doi.org/10.3390/met10030329